3316 | Phys. Chem. Chem. Phys., 2015, 17, 3316--3325 This journal is© the Owner Societies 2015

Cite this:Phys.Chem.Chem.Phys.,

2015, 17, 3316

Ab initio multiple cloning simulations of pyrrolephotodissociation: TKER spectra and velocity mapimaging†

Dmitry V. Makhov,a Kenichiro Saita,‡a Todd J. Martinezbc and Dmitrii V. Shalashilina

We report a detailed computational simulation of the photodissociation of pyrrole using the ab initio

Multiple Cloning (AIMC) method implemented within MOLPRO. The efficiency of the AIMC

implementation, employing train basis sets, linear approximation for matrix elements, and Ehrenfest

configuration cloning, allows us to accumulate significant statistics. We calculate and analyze the total

kinetic energy release (TKER) spectrum and Velocity Map Imaging (VMI) of pyrrole and compare the

results directly with experimental measurements. Both the TKER spectrum and the structure of the

velocity map image (VMI) are well reproduced. Previously, it has been assumed that the isotropic

component of the VMI arises from long time statistical dissociation. Instead, our simulations suggest that

ultrafast dynamics contributes significantly to both low and high energy portions of the TKER spectrum.

I. Introduction

The dynamics of electronically nonadiabatic reactions is key tounderstanding many processes in chemistry and biochemistry.For example light harvesting in plants, fluorescence of livingorganisms, and visual reception all involve photochemicalreactions that include electronic excitation and subsequentelectronically nonadiabatic dynamics. Recently significantprogress has been made in experimental ultrafast time resolvedspectroscopy studies of various photochemical reactions,focused on biologically related molecules. The derivatives ofheteroaromatic molecules such as pyrrole, imidazole, and phenolare important chromophores of nucleobases and aromatic aminoacids. The mechanisms of their photochemistry have been a focusof experimental1 and theoretical2 attention. It was suggested thatthe N–H/O–H bond fission was an important channel in thephotodissociation dynamics and the role of the 1ps* states in thisprocess has been emphasized.3 Total kinetic energy release(TKER) spectra and velocity map images (VMI) for various bondfission reactions were reported providing invaluable informationbut to the best of our knowledge VMI and TKER spectra have

never been calculated theoretically for these molecules. This isa difficult task because TKER spectra reflect important detailsof quantum dynamics in multidimensional systems, whererealistic calculations beyond simple reduced dimensionalitymodels are challenging.

In particular, the photodissociation of pyrrole (C4NH5), proto-typical of many heteroaromatic systems, has been investiga-ted.1a,4 N–H bond fission is the primary decay mechanismfor pyrrole molecules excited to the ps* state. Dissociating Hatoms have been detected5 with a centre-of-mass frame energy of18 kcal mol�1 and a strong anisotropy, indicating promptdissociation (within the molecular rotation period). These initialfindings were supported by further evidence from ion-imaging1b

and the hydrogen (Rydberg) atom photofragment translationalspectroscopy (HRAPTS)4a experiments. The high resolutionHRAPTS data showed several notable features, including direct(Herzberg–Teller allowed) excitation into the 1A2 electronic state,with preservation of the H–T active vibrations into the productradical. Time resolved studies have also been performed6,7 albeitin the region of the stronger1 absorption.

Previous quantum dynamics calculations1b,2c,d have been per-formed using reduced dimensionality model PESs. These havepredicted rapid dissociation and significant population of the 2B1

state of pyrrolyl, which is difficult to reconcile with the experi-mental data. Theoretical work by Sobolewski and Domcke2a,b

showed the existence of conical intersections in the decaychannel, which are by now well-known and opened the possi-bility of electronic population branching during the dissocia-tion. Despite the fact that both 2A2 and 2B1 states of pyrrolyl areenergetically accessible, the interpretation of the HRAPTS data

a Department of Chemistry, University of Leeds LS2 9JT, UK.

E-mail: D.Makhov@leeds.ac.uk, D.Shalashilin@leeds.ac.ukb Department of Chemistry and The PULSE Institute, Stanford University, Stanford,

CA 94305, USA. E-mail: toddjmartinez@gmail.comc SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c4cp04571h‡ Current address: School of Chemistry, University of Edinburgh, West MainsRoad, EH9 3JJ Edinburgh, Scotland, United Kingdom.

Received 9th October 2014,Accepted 9th December 2014

DOI: 10.1039/c4cp04571h

www.rsc.org/pccp

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by experimentalists is that population was only observed in thelowest (X2A2) electronic state of pyrrolyl. The peaks in the TKERspectrum were accordingly assigned to various vibrationalstates of this ground electronic state of pyrrolyl radical.

Recently we performed our first simulation of the dynamicsof photodissociation of pyrrole8 using the MulticonfigurationalEhrenfest (MCE)9 method. MCE uses Ehrenfest trajectoriesdressed with Gaussian wave packets, which serve as a basisset for quantum propagation of the nuclear wave packet. Eachof these trajectory basis functions (TBFs) has an associatedcomplex time-dependent amplitude and they are all coupled toeach other (through the nuclear Schrodinger equation). Thus,the MCE method is fully quantum mechanical in principle.However, the underlying equations of motion in the Ehrenfestmethod can be problematic in regions where electronic statesare decoupled. Detailed balance is not obeyed10 and the potentialenergy surfaces governing the propagation are averages ofthe adiabatic states.11 With a sufficiently large basis set, theseproblems would be corrected by the solution of the time-dependent Schrodinger equation for the amplitudes of the basisfunctions. However, this could require an impractical number ofbasis functions. We recently showed12 how these problems canbe overcome by introducing cloning of Ehrenfest configurations,analogous to the spawning procedure for adaptive basis setexpansion used in ab initio multiple spawning (AIMS).13

The ab initio multiple cloning (AIMC) method is based onMCE dynamics including cloning and coupled to an electronicstructure package to calculate the potential energy surfaces,gradients, and nonadiabatic coupling matrix elements ‘‘on thefly’’ as needed by the dynamics. The dynamics calculations arecarried out in full dimensionality, i.e. all degrees of freedom areincluded explicitly. Similarly to AIMS, the AIMC method uses abasis of trajectory guided Gaussian wave packets to representthe motion of the nuclei.

Although AIMC differs from AIMS in the equations ofmotion for the TBFs, the methods are nevertheless similarenough to be implemented within the same code, and bothmethods have been implemented in MolPro. Recently wedeveloped a new implementation of the MCE method withinthe AIMS code, which combines the best features of the twomethods. Our ab initio Multiple Cloning (AIMC) approach12 hasa number of useful features:

(1) The trajectory basis functions are propagated usingEhrenfest equations of motion. These can be quite accuratein regions where two or more electronic states are stronglycoupled and may then be preferable to classical equations ofmotion on a specific electronic state.

(2) As discussed above, the Ehrenfest equations of motioncan be problematic after passing a strong coupling region.There Ehrenfest trajectories, which are guided by potentialenergy surface average of those of individual electronic states,may move outside of the dynamically important region. This isremedied in the AIMC method by a procedure called cloning,which is an adaptation of spawning procedure of AIMS.Cloning reprojects the Ehrenfest trajectories on the individualelectronic states after quantum transitions are completed.

Importantly, this projection is done in a manner that does notalter the nuclear wavefunction at the time of cloning (similarto spawning in AIMS). The AIMC method combines theadvantages of the AIMS and MCE methods.

(3) AIMC uses an interpolation procedure for calculating matrixelements that is based solely on quantities calculated at the centersof the TBFs. There are no calculations of energies, gradients, ornonadiabatic coupling matrix elements at geometries intermediatebetween trajectory centers. This minimizes the number of expen-sive ab initio electronic structure calculations required during thedynamics. Additionally, it is possible to calculate many of theunderlying Ehrenfest trajectories independently and then to laterrecombine them in a ‘‘post-processing’’ procedure. The quantumamplitudes are calculated in this post-processing stage. We refer tothis as ‘‘incremental propagation’’ below and it has the advantagethat quantum mechanical coupling can be computed at almost nocost in comparison to the ab initio ‘‘on the fly’’ calculation of theensemble of Ehrenfest configuration TBFs.

(4) AIMC makes use of the idea of train basis sets14 (alsoknown as time-displaced basis functions15). Train basis func-tions follow each other along the same Ehrenfest trajectory butwith a time delay, so that propagating trains does not requireadditional electronic structure ab initio information. Trainbasis sets reduce the noise and improve the quality of quantumdynamics calculations at almost no extra cost.

Like other ab initio methods, such as Multiple Spawning13b–e

and Variational Gaussians,16 AIMC simulates the dynamics infull dimensionality and without precomputed potential energysurfaces. Potential energy surfaces and their coupling matrixelements are calculated ‘‘on the fly’’ by solving the electronicSchrodinger equation. The selection of a few important degreesof freedom for the dynamics is avoided, in some cases allowingone to see that unexpected coordinates are important. AIMC(and also AIMS and previous ab initio MCE simulations) are notbiased by the choice of which coordinates to model, since all areincluded on equal footing. The ab initio MCE method (withoutcloning) has been used previously for a more complete analysis ofthe dissociation of pyrrole.8 This showed that the 2B1 state of thepyrrolyl radical is only weakly populated, confirming the originalassignment of the HRAPTS data by experimentalists. We noticedthat after partial departure of the H-atom, the two electronicstates 2A2 and 2B1 are still coupled via an intersection between thepotential energy surfaces of the pyrrolyl radical, which is nearlyisolated. This intersection is reached by the pyrrolyl vibrationalmotion, and as a result the population of the higher 2B1 stateresulting from rapid nonadiabatic dynamics of the N–H bondfission leaks to the lower 2A2 state via longer time dynamics atthis final intersection. A possible role of this feature has beenproposed previously,2c but a multidimensional simulation showsit clearly. Previously, such a conical intersection between twolowest electronic states of pyrrolyl has been found17 and ourcurrent simulation suggests that it may play an important role inthe nonadiabatic dynamics of pyrrole.

In this paper we report a more detailed simulation of pyrroledissociation after excitation to the first excited state, whichcorresponds precisely to the conditions of the recent experiment1a

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3318 | Phys. Chem. Chem. Phys., 2015, 17, 3316--3325 This journal is© the Owner Societies 2015

where the TKER spectrum was measured after 250 nm excitationof pyrrole. The experiment1a uses a 120 fs pump pulse, creating awave packet in the Franck–Condon region, which is convenientlymodeled with AIMC. The experiment also exploits the fact that thetransition dipole moment for the chosen excitation can be wellapproximated as perpendicular to the N–H bond in pyrrole. Thus,the angular distribution of dissociated H atoms with respect tothe pump pulse (velocity map imaging or VMI) provides informa-tion about the detailed reaction dynamics. Our AIMC calculationreproduces very well both the TKER spectrum and the stronganisotropy of the VMI observed in recent experiments,1a validatingthe mechanistic information inferred from the simulations.

II. TheoryII.1 Working equations

We start with a brief sketch of the MCE method and itsimplementation, as full details have been provided previously.12

In the Multiconfigurational Ehrenfest (MCE) approach, the wave-function is represented in a basis set of TBFs |cn(t)i composed ofnuclear and electronic parts:

cnðtÞj i ¼ wnðtÞj iXI

aðnÞI ðtÞ jIj i

!; (1)

where the electronic part is a superposition of several electroniceigenfunctions |jIi, and the nuclear part |wn(t)i is a Gaussianfunction moving along an Ehrenfest trajectory:

wn R;Rn;Pn

� �

¼ 2ap

� �Ndof=4

exp �a R� Rn

� �2þ i

�hPn R� Rn

� �þ i

�hgnðtÞ

� �(2)

The parameter a here determines the width of the Gaussians,which for example can be taken according to previous prescrip-tions.18 %Rn(t) and %Pn(t) are the phase space coordinate andmomentum vectors of the n-th basis function center, and gn isa phase which evolves as:

dgndt¼ Pn R

�

n

2: (3)

The evolution of the Ehrenfest amplitudes a(n)I is driven by the

equations

_aðnÞI ¼ �

i

�h

XJ

HelðnÞIJ a

ðnÞJ ; (4)

where matrix elements of the electronic Hamiltonian Hel(n)IJ are

expressed as:

HelðnÞIJ ¼

VI�Rnð Þ; I ¼ J

�i�h�PnM�1dIJ �Rnð Þ; IaJ

(: (5)

Here VI( %Rn) is the Ith potential energy surface and dIJ( %Rn) =hjI|rR|jJi is the Nonadiabatic Coupling Matrix Element(NACME), and M is the diagonal matrix of nuclear masses.

The centers of the Gaussian basis functions follow Newton’sequations of motion:

�R�n ¼ M�1�Pn

�P�n ¼ �Fn

(6)

where the force %Fn includes both the usual gradient term andalso the Hellmann–Feynman force20 related to nonadiabaticcoupling:

�Fn ¼XI

aðnÞI

��� ���2rRVI�Rnð Þ

þXIaJ

aðnÞI

� ��aðnÞJ dIJ �Rnð Þ VI

�Rnð Þ � VJ�Rnð Þ½ �:

(7)

Eqn (4)–(7) form a complete set of equations determining theevolution of the Ehrenfest TBFs |cn(t)i.

A single Ehrenfest configuration is not flexible enough toaccurately describe quantum dynamics. In the MCE approach,multiple Ehrenfest configurations are used and these representa basis set in which the total wavefunction is expanded as:

jCðtÞi ¼Xn

cnðtÞ cnðtÞj i: (8)

The evolution of the amplitudes cn(t) are determined by:

Xn

cmðtÞ j cnðtÞh i _cnðtÞ ¼ �i

�h

Xn

Hmn � i�h cmðtÞd

dt

��������cnðtÞ

� �cnðtÞ;

(9)

obtained simply by substituting the wave function (8) into thetime dependent Schrodinger Equation, where

Hmn ¼XIJ

aðmÞI

� ��aðnÞJ wmjIh jH wmjIj i: (10)

The term cmðtÞd

dt

��������cnðtÞ

originates from time dependence of

the TBFs and is given by:

cm

���� dcn

dt

¼ wm

���� dwndt

XI

aðmÞI

� ��aðnÞJ

þ wm j wnh iXI

aðmÞI

� ��_aðnÞJ ;

(11)

where

wm

���� dwndt

¼ �R

�n wm

d

d�Rn

��������wn

þ �P�n wm

d

d�Pn

��������wn

� �

þ i

�h_gn wm j wnh i:

(12)

The expressions for matrix elements entering eqn (11) and (12)and complete derivation of all MCE equations have been givenpreviously.12

II.2 Matrix element approximations

Assuming that the electronic wave function depends weaklyon R, so that the second derivative with respect to R can be

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disregarded, the matrix elements of the Hamiltonian can bewritten as

wmjIh jH wnjJj i ¼ dIJ wm ��h2

2rRM

�1rR þ VI ðRÞ����

����wn

� �h2 wmh jdIJðRÞM�1rR wnj i:

(13)

The matrix element of the kinetic energy operator

wm ��h2

2rRM

�1rR

��������wn

is easily calculated analytically. MCE

makes some very simple but reasonable approximations for thematrix elements of the potential energy and nonadiabaticcoupling, as described previously:12

wmh jVI ðRÞ wnj i

� wm j wnh i VI�Rnð Þ þ VI

�Rmð Þ2

� �

þ wmh j R� �Rnð Þ wnj irRVI�Rnð Þ þ wmh j R� �Rmð Þ wnj irRVI

�Rmð Þ2

� �(14)

wmh jdIJðRÞM�1rR wnj i

¼ i

2�hwm j wnh i �PmM

�1dIJ �Rmð Þ�

þ �PnM�1dIJ �Rnð Þ

�:

(15)

In short, the required off-diagonal (in the trajectory index)matrix elements are given by interpolation from the matrixelements along the TBFs (which are needed for propagation ofthe TBFs). With these matrix element approximations, theequations for the amplitudes of the Ehrenfest TBFs, eqn (9),can be evaluated using the same information needed to propa-gate the TBFs. Thus quantum coupling between the configura-tions comes at almost no extra cost. Moreover, eqn (9) can besolved after the trajectories of the Ehrenfest TBFs have beenevaluated (provided all the electronic structure informationused to propagate the TBFs has been saved).

II.3 Basis sets: cloning, trains, and incremental propagation

The Ehrenfest basis set is guided by an average potential whichcan be advantageous when quantum nonadiabatic coupling isstrong and quantum transitions are frequent, but it also becomesunphysical when two or more electronic states have significantamplitudes and very different shape (leading to wave packetbranching after leaving the nonadiabatic coupling region).

In order to reproduce the branching of the wave function, weadopted the spawning procedure from MS method into the socalled Ehrenfest configuration cloning,12 where a basis func-tion is replaced by two functions, each guided by a singlepotential energy surface. After cloning, an Ehrenfest configu-

ration cnj i ¼ wnj iPJ

aðnÞJ jJj i

� �yields two configurations:

c0n�� �

¼ wnj iaðnÞI

aðnÞI

��� ���� jIj i þXJaI

0� jJj i

0B@

1CA (16)

and

cn00

��� E¼ wnj i 0� jIj i þ

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� a

ðnÞI

��� ���2r X

JaI

aðnÞJ jJj i

0BB@

1CCA: (17)

The first clone configuration has nonzero amplitudes for onlyone electronic state, and the second clone contains contribu-tions of all other electronic states. The amplitudes of the twonew configurations become:

cn0 ¼ cn a

ðnÞI

��� ���; cn00 ¼ cn

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� a

ðnÞI

��� ���2r

; (18)

so that the contribution of the two clones |cn0i and |cn0

0i to thewhole wave function (2) remains the same:

cn|cni = cn0|cn

0i + cn00|cn0

0i. (19)

We define a ‘‘breaking force’’ for each state in anEhrenfest TBF:

FðbrÞI ¼ aIj j2 rVI �

XJ

aJj j2rVJ

!(20)

which is the force pulling the I-th state away from the remainingstates. The cloning procedure is applied when the breakingforce is sufficiently large, and, at the same time, the non-adiabatic interstate coupling is small. The second condition isimportant to avoid applying cloning when extensive populationtransfer between electronic states is taking place. Thus, cloningis often applied shortly after a trajectory passes near a conicalintersection.

The solution of the Schrodinger eqn (9) was performed usingcoherent state train basis sets,13a,14 where Gaussian basis func-tions are moving along the same Ehrenfest trajectory but with atime-shift Dt. This approach ensures the preservation of theinteraction between basis-functions during the run, and the sameelectronic structure data are used repeatedly by all basis-functionsin the train drastically reducing the computational cost.

We also have used the incremental propagation, also referredto as ‘‘bit-by-bit’’ propagation.9c In this procedure, the initialwavefunction is expanded as a superposition of Gaussian wave-packets. Subsets of these Gaussian basis functions (the ‘‘bits’’)are then propagated independently. First, each ‘‘bit’’ gives anorigin to a trajectory, which can branch by cloning. Then, eachtrajectory defines a single train basis for propagating the eqn (9).After all trajectories describing the initial ‘‘bits’’ of the wavefunc-tion have been calculated with the help of eqn (4) (includingcloning) and the basis trains have been set, a post-processingstep solves the eqn (9) for the coupled amplitudes of the of trainbasis functions in each ‘‘bit’’. The matrix element approxima-tions discussed in Section II.2 make it efficient and convenientto do this as a post-processing step because all of the neededelectronic structure information can be stored and does notneed to be recalculated. Finally, the contribution of the ‘‘bits’’is properly averaged. Although the ‘‘bits’’ are not coupled witheach other, incremental ‘‘bit-by-bit’’ propagation is not an

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approximation but rather a convenient way to improvesampling, replacing propagation of the whole wave functionwith a large basis by many smaller basis set propagations.‘‘Bit-by-bit’’ approach is accurate provided that each ‘‘bit’’ ispropagated exactly, which is true on a short time scale. Fig. 1sketches all sampling ideas used in the AIMC approach. Thebits of the initial wave function shown by blue circles generatebasis set trains, which are propagated and cloned afterpassing the intersections.

Apart from trivial differences in the notations, AIMC12

differs from ab initio MCE8,9 in the introduction of cloning,time displaced basis sets or trains, and a more accurateestimation (14) of the matrix element of the potential energy.On the other hand in the ref. 8 and 9 more than one trajectorywas originated from each ‘‘bit’’ providing the basis set for thepropagation of eqn (9). In this work only one trajectory per‘‘bit’’ was used to provide a train basis set but in principle theuse of more than one train per ‘‘bit’’ with coupling between thetrains is possible, although at higher computational cost.

III. Computational details

In previous TKER experiments on pyrrole,3a,4a,19 a long reso-nance pulse was used, creating a narrow band of electronicallyexcited vibrational states. In contrast, the most recent experi-ment1a uses a short femtosecond pulse and creates a Franck–Condon vibrational wave packet, which is broad in energy butlocalized in phase space. As these initial conditions are closelyrelated to the TBFs in the AIMC approach, this femtosecondexperiment is well suited for simulation with AIMC.

Using our AIMC approach, we simulated the dynamics ofpyrrole following excitation to the first excited state. Trajec-tories were calculated using AIMS-MOLPRO,14 which was modi-fied to incorporate Ehrenfest dynamics. Electronic structurecalculations were performed with the complete active spaceself-consistent field (CASSCF) method using the cc-pVDZ basis

set. As in previous work,8 we used an active space of eight electronsin seven orbitals (three ring p orbitals and two corresponding p*orbitals, one s orbital and a corresponding s* orbital). Stateaveraging was performed over four singlet states using equalweights, i.e. the electronic wavefunction is SA4-CAS(8,7)/cc-pVDZ.The width of Gaussian functions a was taken as 4.7 Bohr�2 forhydrogen, 22.7 Bohr�2 for carbon, and 19.0 Bohr�2 for nitrogenatoms, as suggested previously.18 Three electronic states weretaken into consideration during the dynamics – the ground stateand the two lowest singlet excited states.

Calculations were run for maximum 200 fs with time-stepB0.06 fs (2.5 a.u.). CASSCF calculations for pyrrole are extre-mely expensive, and each trajectory requires around a month ofCPU time. In order to reduce computational time, calculationsfor the majority of trajectories were stopped as soon as N–Hdistance exceeded 3 Å, excluding a small number of trajectoriesfor which simulations exhibiting N–H dissociation were carriedout to a full 200 fs in order to investigate the post-dissociationdynamics of pyrrolyl radical. Although this limitation makesimpossible proper adiabatic state population analysis, it doesnot affect TKER spectrum as the interaction between departedH-atom and the radical is extremely small at 3 Å distance.

The initial Ehrenfest configurations were randomly sampledfrom the ground state vibrational Wigner distribution in theharmonic approximation. The transition from the ground tothe first excited state is symmetry-forbidden in the Franck–Condon approximation and only occurs due to the coordinate-dependence of the transition dipole moment. In this work, weapproximate the photoexcitation by simply lifting the groundstate wavepacket to the excited state, as would be appropriatefor an instantaneous excitation pulse within the Condonapproximation. Of course, the fine details of the initial photo-excited wavepacket are not completely accounted for in thisapproximation (which assumes the transition dipole moment forthe transition is finite and independent of nuclear coordinates).We do not expect these details to have much effect on theobservables shown in this paper, but future work can verify thisby including the coordinate dependence of the transition dipolemoment in the description of the excited state wavepacket.

Another interesting feature of the experiment1a is that itlargely relies on the fact that only the components of the fieldwhich are perpendicular to the N–H bond can excite the A2

electronic state of pyrrole. See ESI† for more details.In these AIMC simulations, we have used 1000 initial ‘‘bits’’

with one initial Ehrenfest configurations per bit. These 1000 initial‘‘bits’’ (and all the configurations they clone) were recombinedaccording to the incremental propagation described above.Cloning was applied to TBFs when the breaking acceleration ofeqn (20) exceeded a threshold of 5 � 10�6 a.u. and the norm ofthe non-adiabatic coupling vector was simultaneously less than2 � 10�3 a.u. After the cloning event, each branch was ascribed aproper weight which was applied in the statistical analysis of theresults. A total of 91 cloning events occurred, leading to 1091 TBFs.Out of this number, 452 trajectories led to the dissociation ofpyrrole molecule; the majority of the remaining 639 trajectoriesdid not reach an intersection within the 200 fs simulation time.

Fig. 1 A sketch of the AIMC propagation scheme. First the wave functionis represented as a superposition of Gaussian Coherent States (‘‘bits’’), eachof which is a central trajectory for a train basis set. After passing theintersection the trains branch in the process of cloning (a similar diagramwould apply to MCE, without the cloning step).

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Cloning contributed significantly to the number of dissociatingtrajectories and their final distribution. We performed the fullAIMC calculation, which included the solution of eqn (9) for longand bifurcating trajectories using the trains made of N = 21Gaussians, separated by 10 time steps. If only one Ehrenfesttrajectory per bit (and therefore only one train per bit) is usedthen the coupling between the basis Ehrenfest configurationswithin the train does not affect the TKER spectrum. The couplingbetween the amplitudes c of Ehrenfest configurations does notcontribute everywhere except the cloning region, where it canresult in some changes in the relative amplitudes and statisticalweight of ‘‘clones’’. Thus, the coupling described by eqn (9) is notvery significant on the occasion of calculating TKER spectrum butstill was taken into account.

IV. Results

In the analysis that follows we take into account the weights ofbranches wn, the sum of which is one for each initial trajectory.Fig. 2 shows the distribution of the dissociation time, which isdefined as the earliest time when the N–H distance exceeds2.5 Å. It can be seen that 35% of trajectories dissociate within200 fs, and most of these (27% of the whole) dissociate withinthe first 50 fs after photoexcitation. We did not find any strikingdifference in the dynamics after passing the intersectionbetween the trajectories that dissociate within 50 fs and thosethat dissociate in the 50–200 fs window.

The kinetic energy distribution of the ejected hydrogen atomis presented in Fig. 3 together with the experimental observationsof Stavros et al.1a Both distributions clearly exhibit two contribu-tions: a large peak at higher energies, and a small contributionat lower energies. While the calculated kinetic energies are onaverage about 1.5 times higher than experimental values,

this difference can be ascribed to a limited accuracy of CASSCFpotential energy surfaces. To evaluate the error introduced byCASSCF PES calculations, we performed multireference perturba-tion theory calculations with multistate corrections (MS-CASPT2)at the Franck–Condon (FC) region of the S1 PES and the asym-ptotic (N–H distance of 3.0 Å) region of the S0 state and noticedthat CASSCF overestimates by 1855 cm�1 the differencebetween the zero point energy corrected potential energies inthese two regions. This means that for a more accurateMS-CASPT2 PES the kinetic energy peak would shift by approxi-mately B1800–1900 cm�1 towards the lower energies improvingsignificantly the agreement with experiment.

The TKER spectrum alone cannot distinguish products inthe ground electronic state with high vibrational energy fromproducts in a higher electronic state with low vibrationalenergy. However, it is easy to make this distinction from ourAIMC simulations. Analysis of the electronic state amplitudesin the Ehrenfest configurations (1) shows that the bifurcationof trajectories while passing through a conical intersectionplays an important role in the formation of a two-peak spec-trum: high kinetic energy product is predominantly in theground state, confirming the conclusion of our previous work,while the low energy peak is formed by mostly low-weightbranches with substantial contribution from excited electronicstates (see the examples of trajectories below). This suggeststhat MCE method (without cloning) would not be able toreproduce correctly the character of TKER spectrum. Performingadditional MCE calculations for pyrrole in order to compare thespectra directly would be too expensive, but our calculations forimidazole, which will be published later, support this suggestion:MCE method indeed gives a TKER spectrum with just one peak,while AIMC calculations correctly reproduce two-peak characterof the spectrum. More detailed information is available from theangular distribution of hydrogen atoms with respect to thedirection of the field. Reproducing this velocity map imagerepresents a challenge for theory, as it is a more sensitiveindicator of the dynamics than the TKER spectrum.

Fig. 2 The distribution of dissociation times for the TBFs, which is definedas the first time when the N–H distance exceeds 2.5 Å.

Fig. 3 Calculated probability distribution of the kinetic energy of hydro-gen atom after the dissociation. The curve is smoothed by replacing delta-functions with Gaussian functions (s = 200 cm�1). The inset shows theexperimentally measured distribution.1a

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Fig. 4 depicts the velocity distribution of hydrogen atomswith respect to the orientation of the molecule; parts (A) and (B)show velocities in the x-direction vs. their projections on the yzplane and in the y-direction vs. xz-projection, respectively. Itcan be seen that high-energy hydrogen atoms travel primarilyin the direction of the N–H bond, while the velocity vectors oflow-energy atoms, although still lying near the xz-plane, aredistributed more or less isotropically in this plane. Comparingthese results with experimental data1a that high-energy atomsare predominantly ejected perpendicular to the laser polariza-tion and the ejection of low-energy atoms is isotropic, we mustconclude that transition dipole moment is orthogonal to theN–H bond for the excitations leading to high-energy ejections,and lies in the xz-plane for the ones leading to low-energyejections. Both conditions are satisfied if the transition dipoleis pointed in x-direction, i.e. orthogonal to the molecule plane.However, we cannot rule out that different mechanisms (withdifferent orientations of transition dipoles) can be responsiblefor high- and low-energy ejections.

In order to calculate the velocity map image with respect tothe laser pulse polarization, we must average the results fromFig. 3 over all possible orientations of the molecule:

I(r, j) p d(r� |v|)Ð Ð Ð

da db dg cos2(x(a, b, g)) d(j� f(y, a, b, g)),(21)

where a, b and g are Euler angles, y is the angle between thevelocity vector v (given by our calculations) and the transition

dipole of the molecule, x(a, b, g) is the angle between thetransition dipole and light polarization vectors, and f(y, a, b, g)is the angle between the light polarization vector and atomvelocity. Here we take into account that the probability ofexcitation is proportional to cos2(x). Integrating over Eulerangles and replacing, as usual, the d-function for |v| with anarrow Gaussian function, we obtain

Iðr;jÞ / exp � r� jvjð Þ2

2s2

!cos2ðyÞ cos2ðjÞ þ 1

2sin2ðyÞ sin2ðjÞ

� �:

(22)

Fig. 5A and B show the simulated velocity map with respectto the laser pulse polarization assuming that the transitiondipole is orthogonal to the N–H bond and directed along x or ydirections respectively. Both frames reproduce well the mainfeature of the velocity map image, which is the anisotropy of theintense high energy part. Frame 5A is also consistent withexperiment1a in the low energy region showing an isotropicdistribution. Again, this may be an indication that the dipolemoment of the transition is predominantly in the x-direction,although admittedly the statistics of both experiment andsimulation are poorer in the region of low energy.

In order to understand the character of this distribution, weplotted several typical trajectories in Fig. 6. Parts (A)–(C) showpotential energy surfaces along trajectory branches, the popula-tions, and N–H distances respectively. Fig. 6-I is an example ofthe trajectory with one cloning event. The dissociation isstarting at about 25 fs time, when the trajectory reaches anintersection for the first time. After passing the intersection,the ground and first excited states are approximately equally

Fig. 4 Calculated velocities of the ejected hydrogen atom with respect tothe orientation of the molecule: velocities in x-direction vs. their projec-tions on the yz plane (A), and in the y-direction vs. xz-projections (B).Definitions of the axes with respect to the molecule are shown.

Fig. 5 Simulated velocity map image with respect to the laser pulsepolarization assuming transition dipole moment pointed in x (A) and y (B)directions. Definitions of x and y axes are given in Fig. 3. The experimentalVMI1a is shown in the inset.

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populated, so the cloning procedure is applied. As one can seefrom Fig. 6-I (A) the potential energy for the ground statebranch remains more or less constant for the rest of the run,while the excited state trajectory practically immediately passesover a potential barrier, which slows down the motion. Thus,the kinetic energies of ejected hydrogen atoms significantlydiffer between two branches: the ground state branch contri-butes to the high energy peak of the distribution in Fig. 2, whilethe excited state branch contributes to the low energy peak. Forground state branch, the remaining vibrational energy of theradical is low, so it remains on the ground state and does notreach the intersection again within the simulation window. Theenergy taken away by the hydrogen atom is lower for the excitedstate branch, leaving the pyrrolyl radical with sufficient energyto pass through numerous intersections and with populationin both ground and excited states. The importance of inter-sections in the radical reached during and after dissociationhas been noticed previously.8 Eventually this will result inquenching to the ground electronic state. However, this does

not affect the TKER spectrum, which only monitors the dynamicsuntil the H atom is lost.

Fig. 6-II shows a more complicated case of a trajectory withthree cloning events. In this example, the trajectory reaches anintersection extremely quickly, at about 5 fs. After the cloning,ground state branch (branch 1) ejects a high-energy hydrogenatom. As in the previous example, this branch remains in theground state and never passes through an intersection again. Theexcited state branch (branch 2) goes to the top of the potentialbarrier, where it reaches an intersection with second excited state.After another cloning, we get two new branches: branch 2–1which is in a first excited state, and branch 2–2 in a secondexcited state. Branch 2–1 quickly leads to dissociation, ejecting alow-energy hydrogen atom. Branch 2–2 passes through two moreintersections; the first intersection causes a practically completetransition back to the first excited state, and the second oneinitiates a third cloning event creating branch 2–2–1 in theground state, and branch 2–2–2 mostly in the first excited state.Branch 2–2–2 leads to dissociation with a low energy hydrogen

Fig. 6 Examples of calculated trajectories with one (I), three (II), and no cloning (III): (A) – potential energy surfaces along the trajectory, (B) – thepopulations of electronic states, (C) – N–H distance. Populations and energies are shown for S0 (blue), S1 (green) and S2 (red) electronic states.

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atom; one can see from the Fig. 6-II (C) that hydrogen speed is alittle bit higher for branch 2–2–2 but still belongs to the low energypeak of the distribution in Fig. 3. For both these branches, theradical keeps sufficient energy after the dissociation and under-goes numerous transitions between the ground, first excited, and,for branch 2–2–2, second excited states. Again, these transitions arenot visible in the TKER spectrum, which only probes the dynamicsuntil H atom dissociation. Branch 2–2–1 remains in the groundstate and does not lead to immediate dissociation; instead weobserve high-amplitudes oscillations of N–H distance.

Fig. 6-III illustrates the simplest possible case – a trajectorywithout cloning. In this particular example, the N-H bondlength oscillates for about 80 fs until trajectory reaches anintersection. As pointed out in Fig. 2, this is unusually long –most of the initial conditions observed to quench to the groundstate (within the 200 fs simulation time) do so within the first20–50 fs. In this case, the hydrogen atom is ejected from theground state with high kinetic energy.

Inspection of the probabilities of electronic states did confirmthe conclusion of our previous paper. Even though the radicalcan be formed in a mixture of electronic states with noticeablecontribution of excited states, it relaxes to the ground electronicstate via the intersection reached by its vibrational motion. How-ever transitions that occur after the departure of the H atom do notaffect the TKER spectrum and velocity map image calculated inthis paper. Only a small fraction of dissociated trajectories retainsignificant contribution of excited states and forms the low kineticenergy part of the TKER spectrum and velocity map image.

V. Summary

In summary we simulated the photodissociation dynamics ofpyrrole excited to the lowest singlet excited state (1A1 - 1A2)using a new efficient implementation of AIMC within the AIMS-MOLPRO code. In our implementation, the computational costis almost the same as classical ‘‘on the fly’’ molecular dynamicseven though the resulting dynamics is substantially quantummechanical. This efficiency allows the accumulation of sufficientstatistics to reproduce the experimental TKER spectrum and theanisotropy of the velocity map images, shedding new light on thedetails of the process and reaffirming our current understandingof the pyrrole photo-dynamics. Currently a number of simula-tions are under way in our group of the molecules such aspyrazole, imidazole previously studied experimentally.6,20

Acknowledgements

DM, KS and DS acknowledge the support from EPSRC throughgrants EP/I014500/1 and EP/J001481/1. TJM acknowledgessupport from the NSF through CHE-11-24515.

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